D-optimal designs for dependent binary variables Daniel Bruce - PowerPoint PPT Presentation

D-optimal designs for dependent binary variables Daniel Bruce daniel.bruce@stat.su.se Department of Statistics, Stockholm University – p. 1/17

What is optimal design ? Experimental design under an optimality criterion, see Atkinson and Donev (1992) No general optimality criterion Criteria based on the information matrix, A-, D-, and E-optimality D-optimality minimizes the determinant of the inverse of the information matrix. Interpreted as the minimum of the confidence ellipsoid of the parameters Discrimination between models, T-optimality – p. 2/17

Advantage and Disadvantage – p. 3/17

Advantage and Disadvantage Advantage Better precision in the parameter estimates, smaller number of required observations with precision maintained – p. 3/17

Advantage and Disadvantage Advantage Better precision in the parameter estimates, smaller number of required observations with precision maintained Disadvantage Optimal designs depend on unknown parameters, which we want to estimate... – p. 3/17

Advantage and Disadvantage Advantage Better precision in the parameter estimates, smaller number of required observations with precision maintained Disadvantage Optimal designs depend on unknown parameters, which we want to estimate... Approaches to handle the parameter dependence – p. 3/17

Advantage and Disadvantage Advantage Better precision in the parameter estimates, smaller number of required observations with precision maintained Disadvantage Optimal designs depend on unknown parameters, which we want to estimate... Approaches to handle the parameter dependence Sequential designs – p. 3/17

Advantage and Disadvantage Advantage Better precision in the parameter estimates, smaller number of required observations with precision maintained Disadvantage Optimal designs depend on unknown parameters, which we want to estimate... Approaches to handle the parameter dependence Sequential designs Optimum on average designs – p. 3/17

Advantage and Disadvantage Advantage Better precision in the parameter estimates, smaller number of required observations with precision maintained Disadvantage Optimal designs depend on unknown parameters, which we want to estimate... Approaches to handle the parameter dependence Sequential designs Optimum on average designs Minimax designs – p. 3/17

The model Two dependent identically distributed binary variables, denoted S 1 and S 2 Complicated dependence structure Identically distributed variables lead to a trinomial model – p. 4/17

The model Two dependent identically distributed binary variables, denoted S 1 and S 2 Complicated dependence structure Identically distributed variables lead to a trinomial model Let S = S 1 + S 2 and P ( S = s ) = π s s = 0 , 1 , 2 for Multivariate generalized linear model (MGLM), see Fahrmeir and Tutz (2001) Response vector ” T “ Y = , Y 1 Y 2 where 8 1 , if S = i < Y i = for i = 1 , 2 . 0 , otherwise : – p. 4/17

The model Expected value »“ ” T – ” T “ µ = E = Y 1 Y 2 π 1 π 2 Link function ” T ” T “ “ ln π 1 ln π 2 = = η, π 1 π 2 π 0 π 0 Linear predictor ” T ” T “ “ η = = = x θ, η 1 η 2 α 1 + β 1 x α 2 + β 2 x where 0 1 @ 1 0 x 0 ” T “ A and θ = x = α 1 α 2 β 1 β 2 0 1 0 x e η 1 e η 2 1 π 0 = π 1 = π 2 = 1 + e η 1 + e η 2 1 + e η 1 + e η 2 1 + e η 1 + e η 2 – p. 5/17

Likelihood, score Likelihood function n n 2 i (1 − π 1 i − π 2 i ) (1 − y 1 i )(1 − y 2 i ) o π y 1 i 1 i π y 2 i L ( θ | y ) = Π i =1 The loglikelihood function n X { y 1 i α 1 + y 2 i α 2 + x i ( y 1 i β 1 + y 2 i β 2 ) − ln (1 + e η 1 i + e η 2 i ) } l ( θ | y ) = i =1 „ ∂η « T „ ∂π « T „ ∂l Score « T u ( θ ) = ∂θ ∂η ∂π which simplifies to 0 1 ( y 1 i − π 1 i ) n B C ( y 2 i − π 2 i ) B C X B C B C x i ( y 1 i − π 1 i ) B C i =1 @ A x i ( y 2 i − π 2 i ) – p. 6/17

Information matrix, criterion function Fisher information h u ( θ ) u T ( θ ) i I ( θ, x ) = E 0 1 π 1 (1 − π 1 ) − π 1 π 2 xπ 1 (1 − π 1 ) − xπ 1 π 2 n B C − π 1 π 2 π 2 (1 − π 2 ) − xπ 1 π 2 xπ 2 (1 − π 2 ) B C X = B C B x 2 π 1 (1 − π 1 ) x 2 π 1 π 2 C xπ 1 (1 − π 1 ) − xπ 1 π 2 B C i =1 @ A − x 2 π 1 π 2 x 2 π 2 (1 − π 2 ) − xπ 1 π 2 xπ 2 (1 − π 2 ) Standardized information matrix M ( θ, x ) = I ( θ, x ) n Criterion function ˛ M − 1 ( θ, ξ ) `˛ ˛ ˛´ ψ { M ( θ, ξ ) } = ln Standardized variance of the predicted response, see Silvey(1980) M − 1 ( θ, ξ ) I ( θ, x ) ˘ ¯ d ( x, ξ ) = tr ∀ xǫ X ξ ∗ is D-optimal iff d ( x, ξ ∗ ) � p ∀ xǫ X – p. 7/17

Examples of the model Plot 1 Plot 2 1 1 P(S=0) P(S=2) P(S=1) 0.8 0.8 P(S=2) 0.6 0.6 0.4 0.4 0.2 0.2 P(S=0) P(S=1) 0 0 −10 −5 0 5 10 15 0 20 40 60 x x Plot 3 Plot 4 1 1 P(S=0) P(S=0) P(S=2) P(S=2) 0.8 0.8 P(S=1) 0.6 0.6 0.4 0.4 0.2 0.2 P(S=1) 0 0 −5 0 5 10 −5 0 5 x x Four examples of different probability distributions for S . The parameters are θ T 1 = ( − 2 , − 9 , 0 . 3 , 1) , θ T 2 = ( − 1 , − 9 , 1 . 1 , 1 . 3) , θ T 3 = ( − 1 , − 5 , 1 , 2) , and θ T 4 = ( − 3 , − 1 , 0 . 5 , 1) – p. 8/17

Examples of the model 4.5 4 d(x, ξ * ) 3.5 3 2.5 2 1.5 P(S=2) P(S=1) 1 0.5 ← P(S=0) 0 −10 0 10 20 30 40 50 60 70 x The probability distribution for S given α 1 = − 1 , α 2 = − 9 , β 1 = 1 . 1 and β 2 = 1 . 3 . The standardized variance of the predicted response for design ξ ∗ , d ( x, ξ ∗ ) 8 9 − 0 . 4719 2 . 3431 32 . 3787 47 . 6213 < = ξ ∗ = 0 . 2514 0 . 2598 0 . 2422 0 . 2466 : ; – p. 8/17

Symmetric model Restriction β 2 = 2 β 1 , gives a model with symmetry properties The log -odds ratio ln Ω = ln 4 + α 2 − 2 α 1 Define x 0 as x 0 = arg max x ∈ X P ( S = 1) . x 0 = − α 2 2 β 1 1 P x = x 0 ( S = 1) = √ 1 + Ω – p. 9/17

Examples of the symmetric model Plot 1 Plot 2 1 1 P(S=0) P(S=2) P(S=0) P(S=2) P(S=1) 0.8 0.8 P(S=1) 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 5 10 −5 0 5 10 x x Plot 3 Plot 4 1 1 P(S=2) P(S=0) P(S=0) P(S=2) 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 ↑ P(S=1) P(S=1) ↓ 0 0 −10 −5 0 5 10 15 −10 −5 0 5 10 15 x x Probability distribution for S as a function of x . The log -odds ratios are ln Ω 1 = − 4 . 61 , ln Ω 2 = − 1 . 61 , ln Ω 3 = 2 . 39 , and ln Ω 4 = 20 . 39 – p. 10/17

Examples of the symmetric model Plot 1 Plot 2 3.1 3 3 2.9 2.5 d(x, ξ * ) d(x, ξ * ) 2.8 2.7 2 2.6 2.5 1.5 0 5 10 −2 0 2 4 6 x x Plot 3 Plot 4 3 3 2.5 2.5 d(x, ξ * ) d(x, ξ * ) 2 2 1.5 1.5 1 1 −10 0 10 20 −10 −5 0 5 10 15 x x d ( x, ξ ∗ ) for the different examples of D -optimal designs – p. 10/17

Examples of the symmetric model Plot 1 Plot 2 3.1 3 3 2.9 2.5 d(x, ξ * ) d(x, ξ * ) 2.8 2.7 2 2.6 2.5 1.5 0 5 10 −2 0 2 4 6 x x Plot 3 Plot 4 3 3 2.5 2.5 d(x, ξ * ) d(x, ξ * ) 2 2 1.5 1.5 1 1 −10 0 10 20 −10 −5 0 5 10 15 x x d ( x, ξ ∗ ) for the different examples of D -optimal designs -4.07 -0.15 4-point design 3-point design 2-point design ln Ω ✲ Number of design points given the log-odds ratio – p. 10/17

Examples of the symmetric model Plot 1 Plot 2 3.1 3 3 2.9 2.5 d(x, ξ * ) d(x, ξ * ) 2.8 2.7 2 2.6 2.5 1.5 0 5 10 −2 0 2 4 6 x x Plot 3 Plot 4 3 3 2.5 2.5 d(x, ξ * ) d(x, ξ * ) 2 2 1.5 1.5 1 1 −10 0 10 20 −10 −5 0 5 10 15 x x d ( x, ξ ∗ ) for the different examples of D -optimal designs -0.15 ✓ ✏ -4.07 4-point design 3-point design ✒ 2-point design ✑ ln Ω ✲ Number of design points given the log-odds ratio – p. 10/17

Examples of the symmetric model Plot 1 Plot 2 3.1 3 3 2.9 2.5 d(x, ξ * ) d(x, ξ * ) 2.8 2.7 2 2.6 2.5 1.5 0 5 10 −2 0 2 4 6 x x Plot 3 Plot 4 3 3 2.5 2.5 d(x, ξ * ) d(x, ξ * ) 2 2 1.5 1.5 1 1 −10 0 10 20 −10 −5 0 5 10 15 x x d ( x, ξ ∗ ) for the different examples of D -optimal designs ✓ ✏ -4.07 -0.15 ✒ 4-point design ✑ 3-point design 2-point design ln Ω ✲ Number of design points given the log-odds ratio – p. 10/17

2-point design Proposed design 8 9 − α 2 − α 2 c c 2 β 1 − 2 β 1 + ξ ∗ = < = β 1 β 1 0 . 5 0 . 5 : ; Assume that β 1 = 1 and α 2 = 0 β 1 = 1 is a scale parameter If α 2 � = 0 a proposed design with the same ln Ω can be derived – p. 11/17